On Homogenization for Piecewise Locally Periodic Operators
Nikita N. Senik · Russian Journal of Mathematical Physics · 2023
We discuss homogenization of a strongly elliptic operator $$\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon_\#) abla$$ on a bounded $$C^{1,1}$$ domain in $$\mathbb R^d$$ with either Dirichlet or Neumann boundary condition. The function $$A$$ is piecewise Lipschitz in the first variable and periodic in the second one, and the function $$\varepsilon_\#$$ is identically equal to $$\varepsilon_i(\varepsilon)$$ on each piece $$\Omega_i$$ , with $$\varepsilon_i(\varepsilon)\to0$$ as $$\varepsilon\to0$$ . For $$\mu$$ in a resolvent set, we show that the resolvent $$(\mathcal A^\varepsilon-\mu)^{-1}$$ converges, as $$\varepsilon\to0$$ , in the operator norm on $$L_2(\Omega)^n$$ to the resolvent $$(\mathcal A^0-\mu)^{-1}$$ of the effective operator at the rate $$\varepsilon_ {\vee} $$ , where $$\varepsilon_ {\vee} $$ stands for the largest of $$\varepsilon_i(\varepsilon)$$ . We also obtain an approximation for the resolvent in the operator norm from $$L_2(\Omega)^n$$ to $$H^1(\Omega)^n$$ with error of order $$\varepsilon_ {\vee} ^{1/2}$$ .